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Solving Systems of Equations in Real-Life Situations

Learn how to solve systems of linear equations and apply them to solve real-life problems involving walkers' meeting points and hiking distances.

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Solving Systems of Equations in Real-Life Situations

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  1. Lesson 5.1

  2. We have studied several linear relations in the previous chapters • Height of elevators • Wind chill • Rope length • Walks in front of a CRB • In this lesson we will consider two or more linear equations together

  3. System of Equations • A system of equations is a set of two or more equations that have variables in common. The common variables relate similar quantities. • You can think of a equation as a condition imposed on one or more variables: • The equation y = 2x imposes the condition that every y value is twice every x value. • A system of equations imposes several conditions simultaneously • The equations y=3x and y = ½(x+1) impose two conditions on y. • Describe the two conditions imposed on y by these two equations.

  4. Where Will They Meet? • In this investigation you will solve a system of simultaneous equations to find the time and distance at which two walkers meet. • Suppose that two people begin walking in the same direction at different average speeds. • 1st walker starts behind the slower 2nd walker • When and where will the faster walker overtake the slower walker?

  5. Setting up the investigation • Mark a 6 m segment at 1 m intervals • Identify • 1st Walker(starts at 0.5 meters and walks toward the 6 m mark at a speed of 1 m/s) • 2nd Walker(starts at the 2 m mark and walks toward the 6 m mark at 0.5 m/s) • A timekeeper (Counts 1 sec. out loud) • A recorder (Records the time and position of each walker separately) • Practice the walks A B 0 6

  6. Collecting Data • When each walker can follow the instruction accurately, record the time and position of each walker and the timer calls out the time.

  7. Writing Equations • Enter the data in the lists of the graphing calculator. • L1 = time • L2 = position of 1st walker • L3 = position of 2nd walker • Write an equation that fits each walkers data • Enter the equations into the graphing calculator • Find the approximate point where the lines intersect. • Explain the real-world meaning of the intersection point • Check that the coordinates of the point of intersection satisfy both of your equations.

  8. What if? • Suppose walker A walks faster than 1 m/s. • How is the graph different? • What happens to the point of intersection? • Suppose that two people walk at the same speed and direction from different starting marks. • What does this graph look like? • What happens to the solution point? • Suppose that two people walk at the same speed in the same direction from the same starting mark. • What does this graph look like? • How many points satisfy this system of equations?

  9. In this Section • You wrote a system of two equations to model a real-world situation. • You solved a system of two equations using tables and graphs. • You found the meaning of the point of intersection in a real-world situation.

  10. Confirming Points of Intersection • Edna leaves the trailhead at dawn to hike 12 mi toward the lake, where her friend Maria is camping. At the same time, Maria starts her hike toward the trailhead, Edna is walking uphill so she averages only 1.5 mi/h, while Maria averages 2.5 mi/hr walking downhill. When and where will they meet? • Define variables for time and distance from the trailhead. • Write a system of two equations to model this situation. • Solve this system by creating a table and finding the values for the variables that make both equations true. Then locate this solution on a graph. • Check your solution and explain its real-world meaning.

  11. Confirming Points of Intersection • Define variables for time and distance from the trailhead. • X = time walked in hours • Y = distance from trailhead in miles

  12. Confirming Points of Intersection • Write a system of two equations to model this situation. • Y=1.5X • Y=12-2.5X

  13. Confirming Points of Intersection • Solve this system by creating a table and finding the values for the variables that make both equations true. Then locate this solution on a graph.

  14. Confirming Points of Intersection • Check your solution and explain its real-world meaning.

  15. Thinking about the exercises • You will be assigned one problem to solve and then present your solution to the group. • P. 276: 1 • P. 277: 2 • P. 277: 3 • P. 277: 4 • P. 278: 6 • P. 279: 11

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